Preprint
Inserted: 5 jul 2026
Last Updated: 5 jul 2026
Year: 2026
Abstract:
We establish a two-mode stability theory for Monge solutions of the multi-marginal optimal transport problem with barycentric quadratic cost. The associated tuple of maps splits into an external barycentric mode and internal relative modes. A quadratic lower bound for the Kantorovich defect controls the internal modes and yields a square-root estimate without invoking any two-marginal stability result. The external mode is the optimal transport map from the fixed source to the Wasserstein barycenter. Combining the resulting two-mode estimate with M\'erigot's sharp stability theorem gives a $\frac{1}{4}$-H\"older estimate for general perturbations, while barycenter-preserving perturbations satisfy a $\frac{1}{2}$-H\"older estimate. We prove that both exponents and the dependence on the weights are optimal. We also examine the scope of such a decomposition beyond the barycentric cost: collective-coordinate perturbations and uniformly concave costs of the sum retain the two-mode estimate, whereas graph interactions, hedonic costs and translation-invariant costs reveal two possible obstructions---loss of relative coercivity and lack of stability for the remaining external modes.
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